{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:38:16Z","timestamp":1776800296523,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"300","license":[{"start":{"date-parts":[[2016,9,14]],"date-time":"2016-09-14T00:00:00Z","timestamp":1473811200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n For integers\n \n \n \n \n k<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n <\/mml:mrow>\n k,n,c<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n k<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n k,n \\ge 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -color Rado number\n \n \n \n \n \n R<\/mml:mi>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R_{k}(n,c)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is defined to be the least integer\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , if it exists or\n \n \n \n \n \u221e\n \n <\/mml:mi>\n \\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n otherwise, such that for every\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -coloring of the set\n \n \n \n \n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n N<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n \\{1,2,\\dots ,N\\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , there exists a monochromatic solution in that set to the equation\n \n \\[\n \n \n \n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n x<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n x<\/mml:mi>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n c<\/mml:mi>\n =<\/mml:mo>\n \n x<\/mml:mi>\n \n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n .<\/mml:mo>\n <\/mml:mrow>\n x_{1}+x_{2}+ \\dots +x_{k}+c =x_{k+1}.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n <\/p>\n

\n In this paper, we mostly restrict to the case\n \n \n \n \n c<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n c \\ge 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and consider two main issues regarding\n \n \n \n \n \n R<\/mml:mi>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R_{k}(n,c)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n : is it finite or infinite, and when finite, what is its value? Very few results are known so far on either one.\n <\/p>\n

\n On the first issue, we formulate a general conjecture, namely that\n \n \n \n \n \n R<\/mml:mi>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R_{k}(n,c)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n should be finite if and only if every divisor\n \n \n \n \n d<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n d \\le n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n \n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n k-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n also divides\n \n \n \n c<\/mml:mi>\n c<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The \u201conly if\u201d part of the conjecture is shown to hold, as well as the \u201cif\u201d part in the cases where either\n \n \n \n \n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n k-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n divides\n \n \n \n c<\/mml:mi>\n c<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , or\n \n \n \n \n n<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n n \\ge k-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , or\n \n \n \n \n k<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 7<\/mml:mn>\n <\/mml:mrow>\n k \\le 7<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , except for two instances to be published separately.\n <\/p>\n

\n On the second issue, we obtain new bounds on\n \n \n \n \n \n R<\/mml:mi>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n n<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R_{k}(n,c)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and determine exact formulae in several new cases, including\n \n \n \n \n \n R<\/mml:mi>\n \n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R_{3}(3,c)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n R<\/mml:mi>\n \n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R_{4}(3,c)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . As for the case\n \n \n \n \n \n R<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n R_{2}(3,c)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , first settled by Schaal in 1995, we provide a new shorter proof.\n <\/p>\n

Finally, the problem is reformulated as a Boolean satisfiability problem, allowing the use of a SAT solver to treat some instances.<\/p>","DOI":"10.1090\/mcom3034","type":"journal-article","created":{"date-parts":[[2015,2,4]],"date-time":"2015-02-04T10:52:50Z","timestamp":1423047170000},"page":"2047-2064","source":"Crossref","is-referenced-by-count":4,"title":["On the \ud835\udc5b-color Rado number for the equation \ud835\udc65\u2081+\ud835\udc65\u2082+\u2026+\ud835\udc65_{\ud835\udc58}+\ud835\udc50=\ud835\udc65_{\ud835\udc58+1}"],"prefix":"10.1090","volume":"85","author":[{"given":"S.","family":"Adhikari","sequence":"first","affiliation":[]},{"given":"L.","family":"Boza","sequence":"additional","affiliation":[]},{"given":"S.","family":"Eliahou","sequence":"additional","affiliation":[]},{"given":"J.","family":"Mar\u00edn","sequence":"additional","affiliation":[]},{"given":"M.","family":"Revuelta","sequence":"additional","affiliation":[]},{"given":"M.","family":"Sanz","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2015,9,14]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"175","DOI":"10.4064\/aa-20-2-175-187","article-title":"A problem of Schur and its generalizations","volume":"20","author":"Abbott, H. 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